Dielectric-constant Gas Thermometry Research Articles

Measurement of pressures up to 7 MPa applying pressure balances for dielectric-constant gas thermometry

For the determination of the Boltzmann constant by dielectric-constant gas thermometry, the uncertainty of pressure measurements in helium up to 7 MPa has been decreased compared with previous achievements (Sabuga 2011 PTB-Mitt. 121 247–55). This was possible by performing comprehensive cross-float experiments with a system of six special pressure balances and the synchronization of their effective areas. It is now possible to measure a helium pressure of 7 MPa with a relative standard uncertainty of 1.0 ppm applying a 2 cm2 piston-cylinder unit, the calibration of which is traceable to the SI base units.

Metrology for Comparison of Displacements at the Picometer Level

An apparatus capable of comparing displacements with picometer accuracy is currently being designed at NIST. In principle, we wish to compare one displacement in vacuum to a second, equal displacement in gas, in order to determine gas refractive index. If the gas is helium, the refractive index is expected to be amenable to high-accuracy ab initio calculations relating refractive index to gas density or to the ratio of pressure and temperature (P/T); the measured refractive index can then be used to infer (P/T) with an accuracy goal of about 1×10-6(relative standard uncertainty). If either the pressure or temperature is known, the refractive index measurement will allow us to determine the second quantity. Our goal is to achieve an uncertainty limited primarily by the uncertainty of the Boltzmann constant (before redefinition of SI units, which will give the Boltzmann constant a defined value). The technique is an optical analog of dielectric constant gas thermometry and can be used in a similar manner. The dimensional metrology is uniquely challenging, requiring picometer-level uncertainty in the comparison of the displacements.

Dielectric-Constant Gas Thermometry and the Relation to the Virial Coefficients

At PTB new dielectric-constant gas thermometry (DCGT) measurements were performed at the temperature of the triple point of water. As discussed recently in an accompanying paper, the main goal was the determination of the Boltzmann constant \(k\) as a contribution to the international efforts directed to a new definition of the base unit kelvin via fixing the value of \(k\). Besides the linear term in the series expansion used for fitting the results of measurements of DCGT isotherms that reveals \(k\), in this paper the higher-order terms are analyzed. For retrieving highly accurate virial coefficients of helium from the data obtained at gas pressures up to 7 MPa, an extended DCGT working equation is developed. Applying this equation, information is deduced on the viral coefficients up to the fourth density virial coefficient. Finally, comparisons with the latest ab initio calculations for the second and third density virial coefficients as well as the second dielectric virial coefficient are performed.

Improved determination of the Boltzmann constant by dielectric-constant gas thermometry

Fellmuth et al (2011 Metrologia 48 382–90) published the first value of the Boltzmann constant k determined by dielectric-constant gas thermometry at the triple point of water (k = 1.380 654 × 10−23 J K−1, standard uncertainty 9.2 parts per million (9.2 ppm)). Since that time, essential progress of this primary thermometry method has been achieved concerning the design and the assembly of the measuring capacitor, the determination of its effective compressibility, the sensitivity of the capacitance bridge, the influence of stray capacitances, the purity of the measuring gas, the pressure measurement, and the scattering and the evaluation of the data. The resulting new k value amounts to k = 1.380 650 9 × 10−23 J K−1 with a standard uncertainty of 4.3 ppm. This value is about 1.5 ppm larger than the CODATA 2010 one, which has a relative uncertainty of 0.9 ppm.

Low-temperature determination of the Boltzmann constant by dielectric-constant gas thermometry

The Boltzmann constant has been determined by performing isotherm measurements around the triple point of neon with dielectric-constant gas thermometry (DCGT). The necessary thermodynamic reference has been established with an uncertainty of about two tenths of a millikelvin, via realizing the International Temperature Scale of 1990, ITS-90, and exploiting the recently published results for the deviation between thermodynamic temperature and the ITS-90. In combination with a detailed treatment of uncertainty estimates and correlations, a value for the Boltzmann constant of kLT = 1.380 657 × 10−23 J K−1 with a standard uncertainty of 15.9 parts per million (ppm) has been determined. This value deviates from the CODATA value, published in 2008, by only 4.7 ppm and is in very good agreement with the value obtained by DCGT at the triple point of water kTPW = 1.380 654 × 10−23 J K−1 with an uncertainty of 9.2 ppm. The weighted mean of kLT and kTPW leads to the DCGT estimate for the Boltzmann constant of k = 1.380 655 × 10−23 J K−1 with an uncertainty of 7.9 ppm.

Determination of the Boltzmann constant by dielectric-constant gas thermometry

Within an international project directed to the new definition of the base unit kelvin, the Boltzmann constant k has been determined by dielectric-constant gas thermometry at PTB. In the pressure range from about 1 MPa to 7 MPa, 11 helium isotherms have been measured at the triple point of water (TPW) by applying a new special experimental setup consisting of a large-volume thermostat, a vacuum-isolated measuring system, stainless-steel 10 pF cylindrical capacitors, an autotransformer ratio capacitance bridge, a high-purity gas-handling system including a mass spectrometer, and traceably calibrated special pressure balances with piston–cylinder assemblies having effective areas of 2 cm2. The value of k has been deduced from the linear, ideal-gas term of an appropriate virial expansion fitted to the combined isotherms. A detailed uncertainty budget has been established by performing Monte Carlo simulations. The main uncertainty components result from the measurement of pressure and capacitance as well as the influence of the effective compressibility of the measuring capacitor and impurities contained in the helium gas. The combination of the results obtained at the TPW (kTPW = 1.380 654 × 10−23 J K−1, relative standard uncertainty 9.2 parts per million) with data measured earlier at low temperatures (21 K to 27 K, kLT = 1.380 657 × 10−23 J K−1, 15.9 parts per million) has yielded a value of k = 1.380 655 × 10−23 J K−1 with uncertainty of 7.9 parts per million.

High-Precision Capacitance Bridge for Dielectric-Constant Gas Thermometry

A high-resolution and high-precision autotransformer ratio capacitance bridge is described that has been built and tested at the Physikalisch-Technische Bundesanstalt for dielectric-constant gas thermometry. Its main component is a homemade high-precision 1 : 1 inductive voltage divider used in a autotransformer configuration. For balancing the bridge, adjustable in-phase and quadrature currents can be injected. A detailed uncertainty budget is presented for measuring small capacitance changes. Considering correlations between main terms in the mathematical model, it is shown that it is possible to measure capacitance changes of at most a few 0.1% with a relative standard uncertainty below one part per million, i.e., with an uncertainty relative to the capacitance value of order one part per billion.

Capabilities for Dielectric-Constant Gas Thermometry in a Special Large-Volume Liquid-Bath Thermostat

Dielectric-constant gas thermometry is being further developed at PTB to measure thermodynamic temperature and the Boltzmann constant at the triple point of water. For this purpose, a huge liquid-bath thermostat with a liquid volume of about 800 L has been built to provide a suitable thermal environment in the central working volume (diameter 0.5 m, height 0.65 m), in which the vacuum chamber containing the measuring system is placed. Measurements of the temperature field in the working volume have been performed with and without the chamber using a mesh of 12 platinum resistance thermometers at appropriate positions. The results verify that the temperature inhomogeneity in the bath is well below 1 mK as necessary. The uncertainty of the temperature measurement inside the measuring system must be of the order of 0.1 mK. Its thermal conditions have been, therefore, investigated in detail, too. Special emphasis was given to the thermalization after temperature changes.

Dielectric-Constant Gas-Thermometry Scale from 2.5 K to 36 K Applying 3He, 4He, and Neon in Different Temperature Ranges

New dielectric-constant gas-thermometry (DCGT) measurements were performed at PTB in the temperature range from 23 K to 36 K. For the first time, besides helium, neon was also used as a measuring gas. The measurements with helium and neon yielded thermodynamic temperature data in the range between 27 K and 36 K. A combination of these data, with former measurements in the range between 2.5 K and 26 K made using 3He and 4He, lead to a temperature scale which coincides with the ITS-90 within a few tenths of a millikelvin even at highest temperatures. The discussion of the uncertainty budget for neon follows the procedures already proved for helium. Furthermore, a comparison between new highly accurate ab initio calculations for the second and third density virial coefficients of helium, and the extended DCGT results are given. For the third virial coefficient of helium, the presented data allow the first experimental check of the new ab initio calculation for the range above 24 K.

Dielectric-Constant Gas-Thermometry Measuring System for the Determination of the Boltzmann Constant at PTB

Dielectric-constant gas thermometry is being further developed at PTB to measure thermodynamic temperature and the Boltzmann constant k at the triple point of water. Due to the small electric susceptibility of gases, the targeted relative uncertainty of k of the order of 2 ppm can be achieved only if gas pressures up to 7 MPa and special 10 pF capacitors for the susceptibility measurements, including very large multi-ring toroidal cross capacitors, are used. This required development of a huge measuring system having a large heat capacity. Since the temperature measurement must be traceable to the triple point of water at a level of the order of 0.1 mK, a corresponding stability and homogeneity of the thermal conditions has to be realized. The design of the system and data characterizing its thermal parameters is described. The experimental results are compared with estimations based both on simple models and finite-element calculations.

Construction and Start-up of a Large-Volume Thermostat for Dielectric-Constant Gas Thermometry

A liquid-bath thermostat with a volume of about 800 L was designed to provide a suitable thermal environment for a dielectric-constant gas thermometer (DCGT) in the range from the triple point of mercury to the melting point of gallium. In the article, results obtained with the unique, huge thermostat without the DCGT measuring chamber are reported to demonstrate the capability of controlling the temperature of very large systems at a metrological level. First tests showed that the bath together with its temperature controller provide a temperature variation of less than ±0.5mK peak-to-peak. This temperature instability could be maintained over a period of several days. In the central working volume (diameter—500mm, height—650mm), in which the vacuum chamber containing the measuring system of the DCGT will be placed later, the temperature inhomogeneity has been demonstrated to be also well below 1mK.

Helium virial coefficients—a comparison between new highly accurate theoretical and experimental data

Critical tests of the accuracy of ab initio theoretical values of the helium virial coefficients have been made based on experiments with a new dielectric-constant gas thermometry (DCGT) setup established at PTB. As the main starting point, most accurate experimental results for the thermodynamic temperature and the dielectric polarizability of 3He and 4He have been deduced recently. This paper now concentrates on the results for the two- and three-particle interaction described by the second and third density virial coefficients of helium. A comparison between experimental DCGT results and new highly accurate theoretical ab initio results shows an impressive agreement. The experimental uncertainties have been estimated via Monte-Carlo simulations and are comparable to the uncertainty of the latest theory.

Radiation Thermometry Toward the Triple Point of Water?

The article describes and evaluates the possibility of using a high-temperature blackbody of accurately known thermodynamic temperature as a reference source for the determination of lower thermodynamic temperatures by spectral radiation thermometry. By applying various intermediate steps, this approach will allow spectral radiation thermometry to be used for the determination of the thermodynamic temperature of the triple point of water with a low uncertainty. The procedure for such an attempt is outlined, theoretical, and practical limits of the resulting uncertainty in thermodynamic temperature are given. The described experimental approach also provides a framework to calculate the uncertainties in determining the thermodynamic temperatures of the defining high-temperature fixed points of the International Temperature Scale of 1990 down to the triple point of water. The estimation of uncertainties is based on current and future values of the relevant contributing components. The uncertainty anticipated in determining the thermodynamic temperature of the triple point of water is 24 mK with current uncertainties and 1.9 mK in the future. Thus, the described approach yields uncertainties that are slightly higher, but comparable to, the tentative uncertainties of other methods—e.g., dielectric constant gas thermometry developed and applied within the framework of the new determination of the Boltzmann constant.

Primary Dielectric-Constant Gas Thermometry in the Range from 2.4 K to 26 K at PTB

New dielectric-constant gas-thermometry (DCGT) measurements were performed at PTB from 2.4 K to 26 K in order to establish a temperature scale with reduced uncertainty. The progress concerning the measurement of capacitance changes, temperature, and pressure compared with the results published in 1996 is described. This is the first step on the way to determine the Boltzmann constant at the triple point of water. At more than 20 temperatures, isotherms were measured and evaluated performing both single- and multi-isotherm fits. Based on this evaluation, a more accurate DCGT scale has been established that is compared with the constant-volume gas-thermometry scale NPL-75, being one basis of the International Temperature Scale of 1990. Coincidence has been found within only a few tenths of a millikelvin above 3.3 K. This gives, together with literature data, confidence with respect to thermodynamic temperature at this level. Emphasis is also given to the results obtained for the virial coefficients, especially below 3.3 K, where the 1996 DCGT results show strong deviations from the expected behavior. The new experimental data for the second virial coefficient are compared with ab initio calculations.

Acoustic thermometry: new results from 273 K to 77 K and progress towards 4 K

We used a quasi-spherical cavity as an acoustic and microwave resonator to measure the thermodynamic temperatures, T, of the triple points of equilibrium hydrogen, neon, argon and mercury and to measure the difference T − T90, in the range 7 K to 273 K. (T90 is the temperature on the International Temperature Scale of 1990 (ITS-90).) In the range 7 K to 24.5 K, our preliminary values of T − T90 agree with recent results from dielectric-constant gas thermometry and achieve uncertainties that are comparable to or smaller than those achievable using the interpolating constant volume gas thermometer as currently defined on the ITS-90. In the range 90 K to 273 K, the present results for T − T90 obtained using a helium-filled, copper-walled, quasi-spherical cavity agree with earlier results obtained using argon-filled, steel-walled or aluminium-walled, spherical cavities. The agreement confirms our understanding of both acoustic and microwave cavity resonators and demonstrates that resonators function as primary thermometers spanning wide temperature ranges. The mutually consistent acoustic thermometry data from several laboratories imply that the values of (T − T90)/T90 are 5 times larger than the uncertainty of T/T90 near 150 K and near 400 K. They also imply that the derivative dT/dT90 is too large by approximately 10−4 near 273.16 K and that dT/dT90 has a discontinuity of 4 × 10−5 at 273.16 K.

Quasi-spherical cavity resonators for metrology based on the relative dielectric permittivity of gases

We evaluate a quasi-spherical, copper, microwave cavity resonator for accurately measuring the relative dielectric permittivity εr(p,T) of helium and argon. In a simple, crude approximation the cavity’s shape is a triaxial ellipsoid with axes of length a,1.001a and 1.005a, with a=5 cm. The unequal axes of the quasi-sphere separated each of the triply degenerate microwave resonance frequencies of a sphere (f11TM,f12TM,…,f11TE,f12TE,…) into three nonoverlapping, easily measured, frequencies. The frequency splittings are consistent with the cavity’s shape, as determined from dimensional measurements. We deduced εr(p,T) of helium and of argon at 289 K and up to 7 MPa from the resonance frequencies flnσ, the resonance half-widths glnσ, and the compressibility of copper. Simultaneous measurements of εr(p,T) with the quasi-spherical resonator and a cross capacitor agreed within 1×10−6 for helium, and for argon they differed by an average of only 1.4×10−6. This small difference is within the stated uncertainty of the capacitance measurements. For helium, the resonator results for εr(p,T) were reproducible over intervals of days with a standard uncertainty of 0.2×10−6, consistent with a temperature irreproducibility of 5 mK. We demonstrate that several properties of quasi-spherical cavity resonators make them well suited to εr(p,T) determinations. Ultimately, a quasi-spherical resonator may improve dielectric constant gas thermometry and realize a proposed pressure standard based on εr(p,T).

Determination of thermodynamic temperature and 4He virial coefficients between 4,2 K and 27,0 K by dielectric constant gas thermometry

Primary dielectric constant gas thermometry (DCGT) has been used to establish a quasi-continuous temperature scale in the range 4,2 K to 27,0 K with a measurement uncertainty which increases with rising temperature from 0,6 mK to 1,2 mK (confidence level 95%). (It should be noted that previous papers refer to the 68% level which yields about one half of the uncertainty values.) The large number of experimental data for 4He (about thirty isotherms, more than 200 triplets of pressure, temperature, and dielectric constant) allowed, furthermore, the determination of the temperature dependencies of the second and third virial coefficients, without any constraints. The virial coefficients thus obtained exhibit good agreement with data based primarily on conventional gas thermometry and potential calculations. The established temperature scale is compared with the isotherm and constant volume gas thermometry scale NPL-75. A flat sinusoidal form for the differences between the two scales may be seen, which is in accordance with other published data. However, the differences are well within the thermodynamic uncertainties estimated for the NPL-75 and the DCGT scale. This result supports the theoretical polarizability value with a relative combined uncertainty of 6 × 10-5, i.e. the bounds for the polarizability value of 4He have been decreased by one order of magnitude compared with the available rigorous theoretical bounds.

Direct Measurements of the Dielectric Virial Coefficients of 4He between 3 K and 18 K

The second and third dielectric virial coefficients of 4He have been measured by the symmetrical gas-expansion technique of Buckingham, Cole and Sutter for six isotherms between 3,00 K and 17,68 K. The effects of helium adsorption are important, and are considered in detail. No temperature dependence of the second coefficient is seen at a level of δb=±0,008 cm3 mol-1, and average values of the second and third coefficients are b = 0,006 (10) cm3 mol-1 and c = -5 (4) cm6 mol-2, respectively. The uncertainty for b is relatively large because it has a systematic dependence on c: reanalysis of all the existing data shows that a more probable value of the third coefficient is c = -2 (1) cm6 mol-2, which when combined with our data gives b (3 K - 18 K) = -0,001 (4) cm-3 mol-1. The very small value suggests that theoretical calculations of the excess polarizability of the helium diatom need to be reconsidered. The small value and the small uncertainty of b indicate that future work to improve experimental values of the second density virial coefficient of 4He for this temperature range may without ambiguity make use of the methods of dielectric constant gas thermometry.

Dielectric Constant Gas Thermometry from 4.2 to 27.1 K

A new method of high-precision isotherm thermometry has been developed in which the density of helium gas is determined from dielectric constant measurements using the virial expansion of the Clausius-Mossotti equation. The technique is shown to be accurate at the level of similar 0.3 mK for the range 4.2-27.1 K, with potential for both increased accuracy and increased range. The results confirm that the gas thermometer scale NPL-75 is internally consistent to better than 1 mK, and that Berry's recent values for the second virial coefficient of helium are correct at the level similar ± 0.05 cm3 mol-1. The results also give a value for the third virial coefficient, C(T) = (5420 ± 225)/T cm6 mol-2, and a value of 517.257 ± 0.025 × 10-3 cm3 mol-1 for the polarizability of 4He.

DIELECTRIC CONSTANT GAS THERMOMETRY AND THE EQUATION OF STATE OF HELIUM

Preliminary measurements are reported of precision Dielectric Constant Gas Thermometry using helium between 4 and 27 K. Values are given for the third density virial coefficient, for the second dielectric virial coefficient, and for the polarizability of helium.